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\(S\)-integral points on elliptic curves – notes on a paper of B. M. M. de Weger. (English) Zbl 1065.11014
B. M. M. de Weger solved the Diophantine equation \(y^2=x^3-228x+848\) (*) [J. Théor. Nombres Bordeaux 9, 281–301 (1997; Zbl 0898.11008)] in \(S\)-integers with \(S=\{2,\infty\}\), by using tools from Algebraic Number Theory and lower estimates for linear forms in complex and \(q\)-adic logarithms of algebraic numbers. In the present paper a shorter solution for the \(S\)-solutions to (*) is given, for a larger set \(S\), namely \(S=\{2,3,5,7,\infty\}\). Now, linear forms in elliptic logarithms are used and the basic tool for computing lower bounds for such linear forms is a theorem of G. Rémond and F. Urfels [J. Number Theory 57, No. 1, 133–160 (1996; Zbl 0853.11055)], which applies to elliptic curves of rank at most 2, hence to the elliptic curve defined by (*). It should be noted that the same authors in cooperation with J. Gebel and H. G. Zimmer have given an alternative approach to (*), which avoids linear forms in \(q\)-adic elliptic logarithms [Math. Proc. Camb. Philos. Soc. 127, No. 3, 383–402 (1999; Zbl 0949.11033)], but the bounds resulting there are much larger than those of the present paper which result from the theorem of Rémond and Urfels.
11D25 Cubic and quartic Diophantine equations
11J86 Linear forms in logarithms; Baker’s method
11G05 Elliptic curves over global fields
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